688 SCIENCE AND HYPOTHESIS 



but that would not prevent them from being rapidly led to remark 

 a certain constant which would be naturally introduced into their 

 equations, and which would be nothing but what we call the area- 

 constant. But then what would happen? If the area-constant is 

 regarded as essential, as dependent upon a law of nature, then in 

 order to calculate the distances of the planets at any given moment it 

 would be sufficient to know the initial values of these distances and 

 those of their first derivatives. From this new point of view, dis- 

 tances will be determined by differential equations of the second order. 

 Woiild this completely satisfy the minds of these astronomers? I 

 think not. In the first place, they would very soon see that in differ- 

 entiating their equations so as to raise them to a higher order, these 

 equations would become much more simple, and they would be espe- 

 cially struck by the difficulty which arises from symmetry. They 

 would have to admit different laws, according as the aggregate of the 

 planets presented the figure of a certain polyhedron or rather of a 

 regular polyhedron, and these consequences can only be escaped by 

 regarding the area-constant as accidental. I have taken this particular 

 example, because I have imagined astronomers who would not be in 

 the least concerned with terrestrial mechanics and whose vision would 

 be bounded by the solar system. But our conclusions apply in all 

 cases. Our universe is more extended than theirs, since we have fixed 

 stars ; but it, too, is very limited, so we might reason on the whole of 

 our universe just as these astronomers do on their solar system. 

 We thus see that we should be definitively led to conclude 

 that the equations which define distances are of an order 

 higher than the second. Why should this alarm us why do we 

 find it perfectly natural that the sequence of phenomena depends on 

 initial values of the first derivatives of these distances, while we 

 hesitate to admit that they may depend on the initial values of the 

 second derivatives? It can only be because of mental habits created 

 in us by the constant study of the generalized principle of inertia 

 and of its consequences. The values of the distances at any given 

 moment depend upon their initial values, on that of their first deriva- 

 tives, and something else. What is that something else? If we do 

 not want it to be merely one of the second derivatives, we have only 

 the choice of hypotheses. Suppose, as is usually done, that this some- 

 thing else is the absolute orientation of the universe in space, or the 

 rapidity with which this orientation varies ; this may be, it certainly is, 

 the most convenient solution for the geometer. But it is not the most 

 satisfactory for the philosopher, because this orientation does not exist. 

 We may assume that this something else is the position or the velocity 

 of some invisible body, and this is what is done by certain persons, 

 who have even called the body Alpha, although we are destined to never 

 know anything about this body except its name. This is an artifice en- 



